POROSITY AND DIFFERENTIABILITY OF LIPSCHITZ MAPS FROM STRATIFIED GROUPS TO BANACH HOMOGENEOUS GROUPS
VALENTINO MAGNANI, ANDREA PINAMONTI, AND GARETH SPEIGHT
Abstract. Let f be a Lipschitz map from a subset A of a stratified group to a Banach homogeneous group. We show that directional derivatives of f act as homogeneous homomorphisms at density points of A outside a σ-porous set. At all density points of A we establish a pointwise characterization of differentiability in terms of directional derivatives. These results naturally lead us to an alternate proof of almost everywhere differentiability of Lipschitz maps from subsets of stratified groups to Banach homogeneous groups satisfying a suitably weakened Radon-Nikodym property.
1. Introduction
Stratified groups are a special class of finite dimensional, connected, simply connected and nilpo-tent Lie groups (Definition 2.4). These groups, equipped with the so-called homogeneous norm, were introduced by Folland in the framework of subelliptic PDE [14]. They subsequently appeared in the work of Pansu under the name of Carnot groups, where their metric structure was defined by the so-called Carnot-Carath´eodory distance [29]. In the last two decades there has been increasing interest in the relationship between the geometry of stratified groups and other areas of mathematics, such as PDE, differential geometry, control theory, geometric measure theory, mathematical finance and robotics [1, 5, 6, 11, 17, 28, 39].
Nevertheless several interesting and challenging problems remain open, since the geometry of stratified groups is highly non-trivial. For instance, rectifiability of the reduced boundary in higher step groups, the isoperimetric problem, the regularity of minimal surfaces, the regularity of geodesics and the validity of general coarea formulae for Lipschitz mappings. One of the basic reasons for these difficulties is that any noncommutative stratified group contains no subset of positive measure that is bi-Lipschitz equivalent to a subset of a Euclidean space [3, 18, 25, 38]. In all of these results, a key role is played by Pansu’s generalization of Rademacher’s theorem to Carnot groups [29].
This differentiation theorem states that every Lipschitz map from an open subset of a stratified group to another stratified group is differentiable almost everywhere with respect to the natural Haar measure. Here differentiability is defined like classical differentiability, but takes into account the new geometric and algebraic structure (Definition 2.18). More broadly, a great effort has been made to understand differentiability properties of Lipschitz functions in different settings, like stratified groups, Banach spaces and more general metric measure spaces. We mention only a few papers [2, 4, 7, 8, 9, 10, 16, 21, 22, 26, 30, 34, 35, 36] to give a glimpse of the many works in this area.
Date: September 11, 2019.
2010 Mathematics Subject Classification. 28A75, 43A80, 49Q15, 53C17.
Key words and phrases. stratified group, Carnot group, Banach homogeneous group, Carnot-Carath´eodory distance, Lipschitz map, differentiability, porous set.
V. M. acknowledges the support of the University of Pisa, (Institutional Research Grant) Project PRA 2016 41 and Project PRA 2018 49. A. P. acknowledges the support of the Istituto Nazionale di Alta Matematica F. Severi.
In the setting of noncommutative stratified groups, the first named author and Rajala extended Pansu’s theorem to Lipschitz mappings on a measurable set of a stratified group and taking values in a Banach homogeneous group (Definition 2.3). They proved the following theorem [27].
Theorem 1.1. Suppose is a Banach homogeneous group whose first layer H1 ⊂ has the
RNP. If f: A → is Lipschitz, A ⊂ and is a stratified group, then f is almost everywhere differentiable.
It is easy to notice that every stratified group is a Banach homogeneous group and the class of finite dimensional Banach homogeneous groups coincides with that of homogeneous groups, as defined in [13]. Commutative Banach homogeneous groups coincide with Banach spaces. The simplest example besides these is an infinite dimensional version of the Heisenberg group H2× . We consider the infinite dimensional real Hilbert space H with scalar product h·, ·i and the group operation is defined as follows:
(h1, h2, t1) · (h01, h02, t01)= (h1+ h10, h2+ h02, t1+ t2+ hh1, h02i − hh2, h01i),
for any (h1, h2, t1), (h01, h02, t01) ∈ H2× . We refer the interested reader to [27] for more examples.
The present paper continues the investigation started in [31], on the relationship between porosity and differentiability of Lipschitz maps on stratified groups. A set in a metric space is (upper) porous (Definition 2.19) if each of its points sees nearby relatively large holes in the set on arbitrarily small scales. A set is σ-porous if it is a countable union of porous sets. Proving a set is σ-porous is useful because σ-porous sets in metric spaces are first category and have measure zero with respect to any doubling measure [40, 41]. Showing a set is σ-porous usually gives a stronger result than showing it is of first category or has measure zero.
It is well known that porosity is a useful tool to study differentiability of Lipschitz mappings. We refer the reader to the survey articles [40, 41] for more information about porous sets and their applications. Porosity has been also recently used in Euclidean spaces to obtain the following result: for any n > 1, there exists a Lebesgue null set in Rncontaining a point of differentiability for every Lipschitz mapping from Rn to Rn−1 [35]. In the Banach space setting, [22, 23] give a version of Rademacher’s theorem for Frech´et differentiability of Lipschitz mappings on Banach spaces in which porous sets are negligible in a suitable sense. Other results have also been studied in stratified groups [20, 30, 32, 33].
In [31], the second and third named authors used porosity to study some fine differentiability properties of Lipschitz mappings from stratified groups to Euclidean spaces. In the present paper, we improve upon [31] in two directions: we consider more general classes of domains and targets of the Lipschitz mapping. We allow the mapping to be defined on a measurable subset of a stratified group. Due to the absence in general of Lipschitz extensions between stratified groups, some technical difficulties appear. The Euclidean space in the target is replaced by a Banach homogeneous group. Thus, the noncommutative infinite dimensional structure of a Banach homogeneous group adds further difficulties.
Let A be a subset of a stratified group , be a Banach homogeneous group, and consider a Lipschitz map f : A ⊂ → . Our first result states that directional derivatives of f at density points of the domain act as homogeneous homomorphisms outside a σ-porous set. Here it is worth to emphasize that the notion of directional derivative (Definition 2.9) takes into account the fact that f need not be defined on a whole neighborhood of the point. For the definition of homogeneous homomorphism, see Definition 2.17. Note that ∂+f(x, ζ) and δaζ denote directional derivatives and
Theorem 1.2. Suppose A ⊂ is measurable and f : A → is Lipschitz. Then there is a σ-porous set P ⊂ such that whenever x ∈ A ∩ D(A) \ P the following implication holds: if ∂+f(x, ζ) and ∂+f(x, η) exist for some ζ, η ∈ , then ∂+f(x, δ
aζδbη) exists for every a, b > 0 and
∂+f(x, δaζδbη) = δa∂+f(x, ζ)δb∂+f(x, η).
Surprisingly, the techniques of [36, Theorem 2] can be extended to the nonlinear framework of stratified groups. Nontrivial obstacles arise from the structure of both the domain and the groups involved. Theorem 1.2 precisely generalizes [31, Theorem 3.7] from real valued Lipschitz mappings to Banach homogeneous group valued Lipschitz mappings.
For real valued Lipschitz mappings, directional derivatives were defined using the Lie algebra of the stratified group and only horizontal directions were used. In the present paper we realize that directional derivatives must be introduced for all directions, using the Lie group structures of and . This important difference is necessary to reflect the noncommutative structure of the target and it was rather unexpected. We notice, indeed, that the restriction of a Lipschitz mapping to a nonhorizontal curve is no longer Lipschitz in the Euclidean sense.
Our second main result characterizes points of differentiability at density points of the domain. For the definition of directional derivatives, see Definition 2.9.
Theorem 1.3. Suppose A ⊂ is measurable, f : A → is Lipschitz and x ∈ A ∩ D(A). Then f is differentiable at x if and only if the following properties hold:
(1) f is differentiable at x in any direction ζ ∈ \ {0}
(2) the mapping 3 ζ → ∂+f(x, ζ) ∈ is a homogeneous homomorphism.
As a consequence of Theorem 1.2 and Theorem 1.3 we prove that, at density points of the do-main except for a σ-porous set, existence of directional derivatives in a spanning set of horizontal directions implies differentiability. If the horizontal space of satisfies the RNP then directional derivatives in this finite set of directions exist at almost every point of A (Theorem 2.16). As a byproduct, taking into account that porous sets have measure zero, we are lead to a different and simpler way to establish both Theorem 1.1 and then also Pansu’s Theorem [29].
The results of this paper illustrate the role played by porosity to pass from almost everywhere existence of horizontal directional derivatives [27, Theorem 3.1] to almost everywhere differentia-bility of Lipschitz maps. The use of porosity provides a more natural approach to differentiability and provides more precise information about how differentiability occurs.
We now describe the structure of the paper. In Section 2 we give the main definitions. In Section 3 we prove Theorem 1.2, while in Section 4 we prove Theorem 1.3. Section 5 is devoted to a few technical facts. The first one is the construction of a separable Banach homogeneous target for our given Lipschitz mapping. We then give a proof of a technical lemma that we use earlier for estimating distances. Finally we prove that the set of points where f is differentiable in some directions is measurable. This measurability is important to establish Theorem 2.16 and its proof is rather technical since our Lipschitz mapping cannot be extended to the whole space.
Acknowledgement. The second and third named authors thank the organizers of the workshop “Singular Phenomena and Singular Geometries”, Pisa, 20-23 June 2016, for the warm hospitality and for the scientific environment, where the project of the present paper started.
2. Preliminaries
2.1. Banach Lie algebras and Banach homogeneous groups. We recall only basic facts about Banach homogeneous groups. More information and additional examples can be found in [27]. Definition 2.1. A Banach Lie algebra is a Banach space equipped with a continuous Lie bracket, namely a continuous, bilinear and skew-symmetric mapping [·, ·] : × −→ that satisfies the Jacobi identity:
[x, [y, z]]+ [y, [z, x]] + [z, [x, y]] = 0 for all x, y, z ∈ .
A Banach Lie algebra is nilpotent if there is ν ∈ N such that whenever x1, x2, . . . , xν+1 ∈ , we
have
[[[· · · [[x1, x2], x3] · · · ], xν], xν+1]= 0
and there exist y1, y2, . . . , yν ∈ such that
[[· · · [[y1, y2], y3] · · · ], yν] , 0.
The integer ν is uniquely defined and is called the step of nilpotence of .
A nilpotent Banach Lie algebra can be equipped with the Lie group operation
(2.1) xy= x + y +
ν
X
m=2
Pm(x, y) ,
which is the truncated Baker-Campbell-Hausdorff series where the Banach Lie bracket is used. For any m ≥ 2, the polynomial Pmabove is given by Dynkin’s formula
(2.2) Pm(x, y)= X(−1)k−1 k m−1 p1!q1! · · · pk!qk! x ◦ · · · ◦ x | {z } p1times ◦ q1times z }| { y ◦ · · · ◦ y ◦ · · · ◦ x ◦ · · · ◦ x | {z } pktimes ◦ qktimes z }| { y ◦ · · · ◦ y.
Here we used the nonassociative product
xi1 ◦ xi2◦ · · · ◦ xik = [[· · · [[xi1, xi2], xi3] · · · ], xik]
and the sum is taken over the 2k-tuples (p1, q1, p2, q2, . . . , pk, qk) such that pi+qi≥ 1 for all positive
i, k ∈ N and Pki=1pi+ qi = m. Note that P2 has the simple form P2(x, y) = [x, y]/2. The explicit
formula for the group product will only be used in the technical Section 5.
Definition 2.2. A nilpotent Banach Lie algebra equipped with the group operation (2.1) is called a Banach nilpotent Lie group.
If S1, S2, . . . , Sn⊂ are closed subspaces of a Banach space such that the mapping
J: S1× · · · × Sn −→ with J(s1, . . . , sn)= n
X
l=1
sl
is a Banach isomorphism (continuous linear bijection), then we write = S1⊕ · · · ⊕ Sn.
Definition 2.3. We say that a Banach nilpotent Lie group of step ν is a Banach homogeneous groupif it admits a stratification. This means there exist ν closed subspaces H1, . . . , Hν such that
= H1⊕ · · · ⊕ Hν,
The stratification equips any Banach homogeneous group with dilations δr: −→ , r > 0.
These are Banach isomorphisms defined by
δr◦πi = riπi
where πi : → Hi is the canonical projection for all i= 1, . . . , ν. These dilations respect the Lie
bracket and Lie group structure:
[δrx, δry]= δr[x, y] and δr(xy)= δr(x)δr(y) for all x, y ∈ , r > 0.
Throughout this article we denote by | · | the underlying Banach space norm on . It can be shown there exist positive constants σ1, . . . , σνwith σ1= 1 such that k · k: → defined by:
(2.3) kx k= max{σi|πi(x)|1/i: 1 ≤ i ≤ ν}
satisfies the standard properties
kδrxk= r kxk and kxyk ≤ kxk + kyk for all x, y ∈ , r > 0.
We say that k · k is a Banach homogeneous norm on . The map ρ(x, y)= kx−1ykis a distance on satisfying
(2.4) ρ(zx, zy) = ρ(x, y) and ρ(δrx, δry)= r ρ(x, y) for all x, y, z ∈ , r > 0.
We say that ρ is a Banach homogeneous distance on . We also write ρ(z)= ρ(z, 0) for z ∈ . An important subclass of Banach homogeneous groups are stratified groups.
Definition 2.4. A stratified group is a Banach homogeneous group for which the underlying Banach space is finite dimensional and whose first layer V1 of the stratification = V1⊕ · · · ⊕ Vs
satisfies
[V1, Vj]= Vj+1for j ≥ 1 and Vj= {0} for j > s.
As a finite dimensional Lie group, is equipped with a Haar measure which we denote by µ. Remark 2.5. While any stratified group is also a finite dimensional Banach homogeneous group, the converse does not hold. To see this it suffices to consider a commutative Banach homogeneous group = H1⊕H2, where H1and H2are finite dimensional vector spaces. Clearly the commutative
Lie product yields [H1, H1]= {0} ⊂ H2, so cannot be a stratified group.
We can equip any stratified group with a homogeneous distance, i.e. a distance satisfying (2.4). We define the integer
Q=
s
X
j=1
jdim(Vj),
the corresponds to the Hausdorff dimension of with respect to any fixed homogeneous distance. It is a standard fact that the Haar measure µ satisfies
µ(xA) = µ(A) and µ(δr(A))= rQµ(A) for measurable A ⊂ , x ∈ and r > 0.
We consequently have
µ(B(x, r)) = rQµ(B(0, 1)) for every open ball B(x, r) ⊂ .
Fixing a basis compatible with the stratification identifies with Rn for a positive integer n. In these coordinates, the Haar measure is simply the n-dimensional Lebesgue measure Ln (up to constant multiplication).
From now on will be a Banach homogeneous group with stratification = H1⊕ · · · ⊕ Hν
equipped with the Banach homogeneous norm k · k and corresponding homogeneous distance ρ. We denote by a stratified group of step s with stratification = V1⊕ · · · ⊕ Vsequipped with a
homogeneous distance d. We also use the notation d(x)= d(x, 0).
2.2. Directional derivatives and differentiability. For functions defined on general measurable domains we may not be able to approach each point from every direction inside the domain. We will introduce an unusual but flexible definition of directional derivative, based on the following approximation of directions inside a domain.
Definition 2.6. Let A ⊂ be measurable and x ∈ . We say that points ζtx ∈ for t > 0
approximate in A the directionζ at the point x if
(2.5) xζxt ∈ A for every t > 0 and d(ζ
t x, δtζ)
t → 0 as t ↓0.
Definition 2.7. Let A ⊂ be a measurable set. We say that x ∈ is a density point of A if lim
r↓0
µ(A ∩ B(x, r)) µ(B(x, r)) = 1. We denote by D(A) the set of density points of A.
Since µ is doubling, the Lebesgue differentiation theorem applies and µ(A \ D(A)) = 0 for any measurable set A ⊂ . It is a standard fact that at every density point x ∈ D(A) we have
(2.6) dist(A, y)
d(x, y) → 0 as y → x.
At density points of a measurable set, we can approximate in every direction.
Lemma 2.8. Suppose A ⊂ is measurable and x ∈ D(A). Then for every ζ ∈ there exist approximating pointsζtx in directionζ at the point x. In addition, we can choose ζxt such that the limit in(2.5) is uniform for 0 < d(ζ) ≤ 1.
Proof. For each t > 0, choose yt∈ A such that
d(yt, xδtζ) < dist(A, xδtζ) + t2d(ζ).
To conclude the proof, apply (2.6) with y= xδtζ and define ζtx= x−1yt.
Definition 2.9. Let A ⊂ be measurable and f : A → . Suppose x ∈ A and ζ ∈ \ {0} for which there exist points which approximate in A the direction ζ at x.
We say that f is differentiable at x in direction ζ with directional derivative ∂+f(x, ζ) ∈ if for
any choice t 7→ ζtxof points approximating in A the direction ζ at x, we have
lim
t↓0δ1/t
f(x)−1f(xζtx) = ∂+f(x, ζ).
Proposition 2.10. Assume the hypotheses of Definition 2.9. Then the directionδaζ can be
approx-imated at x for every a> 0. Further, if ∂+f(x, ζ) exists then also ∂+f(x, δaζ) exists for every a > 0
and we have∂+f(x, δaζ) = δa∂+f(x, ζ).
Proof. Let ζtxapproximate the direction ζ at x. Then xζatx ∈ A and
d(ζatx, δtδaζ)
t =
ad(ζxat, δatζ)
Hence ζatx approximates the direction δaζ at x.
Now suppose ηtx approximates the direction δaζ at x. By the above argument, ηt/ax approximates
the direction ζ at x. Using differentiability in direction ζ gives lim t↓0 δ1/t( f (x) −1f(xηt x))= δalim t↓0 δ1/t( f (x) −1f(xηt/a x ))= δa∂+f(x, ζ). Hence ∂+f(x, δaζ) = δa∂+f(x, ζ).
Remark 2.11. Let A ⊂ be measurable, f : A → be Lipschitz and x ∈ A. Suppose there exists one curve ζtx ∈ approximating direction ζ at x with
lim
t↓0 δ1/t
f(x)−1f(xζtx) = w ∈ .
Then for any other curve ηtxapproximating direction ζ at x, we also have
lim t↓0δ1/t f(x)−1f(xηtx) = lim t↓0 δ1/t f(x)−1f(xζxt) = w ∈ .
In other words, f is differentiable at x in direction ζ and w = ∂+f(x, ζ).
Remark 2.12. In the simple case where A ⊂ is open, we may choose the simplest curve ζxt = δtζ
for sufficiently small t. Hence, by Remark 2.11, to verify directional differentiability for a Lipschitz function f : A → , we have only to check the existence of the limit
lim
t↓0 δ1/t
f(x)−1f(xδtζ) = ∂+f(x, ζ).
Since for Lipschitz functions directional differentiability requires only one curve ζt
x, it is useful
to find those points of the domain where we have a natural choice.
Definition 2.13. Let A ⊂ and x, ζ ∈ . We say that A is dense at x in direction ζ if L1({0 < θ < t : xδθζ < A})
t → 0 as t ↓ 0.
Proposition 2.14. Suppose A ⊂ is measurable and dense at x ∈ A in directionζ. Let f : A → be Lipschitz. Then the following are true:
(1) there exist ζtxapproximating in A the directionζ at x
(2) defining A(x, ζ)= {t ∈ R | xδtζ ∈ A}, we have 0 ∈ A(x, ζ)
(3) f is differentiable at x in direction ζ if and only if the following limit exists: lim
t↓0, t∈A(x,ζ)δ1/t( f (x) −1f(xδ
tζ)).
(2.7)
In either case, the above limit equals∂+f(x, ζ). Proof. For t > 0, choose 0 < T (t) ≤ t such that xδT(t)ζ ∈ A and
t − T(t) < inft − a : 0 < a ≤ t, xδaζ ∈ A + t2.
Density of A at x in direction ζ implies T (t)/t → 1. The curve defined by ζtx = δT(t)ζ then satisfies
property (1).
Property (2) is clear from the definition of directional density.
Suppose now that the limit (2.7) exists. Choosing the previously defined curve ζtx, this gives the
existence of the limit lim t↓0δ1t( f (x) −1f(xζt x))= lim t↓0 δT(t)t δ 1 T(t)( f (x) −1f(xδ T(t)ζ)) .
By Remark 2.11, f is differentiable at x in direction ζ with ∂+f(x, ζ) equal to the limit in (2.7). Conversely, assume that f is differentiable at x in direction ζ. Considering again the previous curve ζxt = δT(t)ζ, it follows that
∂+f(x, ζ)= lim t↓0, t∈A(x,ζ)δ1/t( f (x) −1f(xζt x))= lim t↓0, t∈A(x,ζ)δ1/t( f (x) −1f(xδ tζ)) Ex,ζ,t, (2.8)
where we clearly have ρ Ex,ζ,t = ρ δ1/t( f (xδtζ)−1f(xζtx)) ≤ L d(δtζ, ζ t x) t → 0.
This proves property (3).
Recall that a Banach space X has the Radon-Nikodym property (RNP) if every Lipschitz map f: [0, 1] → X is differentiable almost everywhere. We say that a subspace of a Banach space has the RNP if it does when considered as a Banach space in its own right. To obtain directional derivatives we use the following theorem [27, Theorem 3.1].
Theorem 2.15. Suppose H1 ⊂ has the RNP. Let A ⊂ andγ : A → be a Lipschitz mapping.
Thenγ is almost everywhere differentiable.
Theorem 2.16. Suppose H1 ⊂ has the RNP. Let A ⊂ be measurable and ζ be horizontal, namelyζ ∈ V1\ {0}. If f : A → is Lipschitz, then for almost every point x ∈ A:
(1) A is dense at x in direction ζ, (2) f is differentiable at x in direction ζ.
Proof. Choose ˜V1⊂ V1such that ˜V1⊕ span{ζ}= V1and consider a basis (e1, . . . , en−1) of
N:= ˜V1⊕ V2⊕ · · · ⊕ Vs,
for which each eibelongs to some subspace Vj. The basis (e1, . . . , en−1, ζ) of allows us to define
Ψ : Rn → as follows Ψ(ξ, t) = n−1 X j=1 ξjej (tζ).
The map Ψ is a global diffeomorphism (see [26, Proposition 7.6]). By 3.1.3(5) of [12], for a.e. (ξ, t) ∈Ψ−1(A) the set of points
{θ ∈ R : (ξ, θ) < Ψ−1(A)}
has density zero at t. Since ζ ∈ V1, henceΨ(ξ, t) = Ψ(ξ, 0)δtζ, the previous statement implies that
for a.e. x= Ψ(ξ, t) ∈ A, the set A is dense at x in direction ζ. We denote the set of these points by Aζ, observing that is measurable and µ(A \ Aζ)= 0.
Let Df,ζ ⊂ Aζ be the set of points at which f is differentiable in direction ζ. Our proof is
completed once we have µ(Aζ\ Df,ζ)= 0, that is LnΨ−1(Aζ\ Df,ζ)
. By the measurability of Df,ζ
(whose proof is technical and postponed to Section 5.3), the set Zf,ζ = Aζ\ Df,ζ is also measurable. We can then apply Fubini’s theorem to the measurable set
Ψ−1(Z f,ζ),
whose measure can be recovered by integration of measures of the 1-dimensional sections Ψ−1(Z
where ξ ∈ Rn−1. The composition t → f (Ψ(ξ, t)) ∈ is Lipschitz on Ψ(ξ, ·)−1(Aζ) ⊂ R, therefore
Theorem 2.15 provides its a.e. differentiability on Ψ(ξ, ·)−1(A
ζ) ⊂ R. It follows that Ψ(ξ, ·)−1(Zf,ζ)
is negligible, hence so isΨ−1(Zf,ζ).
Definition 2.17. A homogeneous homomorphism, in short h-homomorphism, from to is a map L: → such that L(xy)= L(x)L(y) and L(δr(x))= δr(L(x)) for all x, y ∈ and r > 0.
Any h-homomorphism is automatically Lipschitz. Indeed, since is stratified there exist a posi-tive integer N and v1, . . . , vN ∈ V1such that for some T > 0 the set
V:= {δt1v1· · ·δtNvN: |ti|< T }
contains the unit ball. The h-homomorphism property then yields the Lipschitz continuity. The reader may also see [24, Proposition 3.11], which is stated there for stratified group targets, but works equally well for Banach group targets.
Definition 2.18. Let A ⊂ and x ∈ A ∩ D(A). We say that f : A −→ is h-differentiable at x, or simply differentiable at x, if there exists an h-homomorphism L: −→ such that
ρ( f (x)−1f(xz), L(z))
d(z) → 0 as z ∈ x
−1Aand d(z) ↓ 0.
The mapping L is unique and called the h-differential of f at x. 2.3. Porous sets. We now define porous sets and σ-porous sets.
Definition 2.19. Let (M, ρ) be a metric space, E ⊂ M and a ∈ M. We say that E is porous at a if there existΛ > 0 and a sequence xn→ a such that
B(xn, Λρ(a, xn)) ∩ E = ∅ for every n ∈ N.
A set E is porous if it is porous at each point a ∈ E withΛ independent of a. A set is σ-porous if it is a countable union of porous sets.
Porous sets in stratified groups have measure zero. This follows from the fact that Haar measure on stratified groups is Ahlfors regular, hence doubling, so the Lebesgue differentiation theorem applies [19, Theorem 1.8].
3. Directional derivatives act as h-homomorphisms outside a σ-porous set
In this section we study applications of porosity to directional derivatives. Recall that is a stratified group of step s and is a Banach homogeneous group of step ν. We will need the following estimate for the Banach homogeneous distance in .
Lemma 3.1. Let N ∈ N and Aj, Bj ∈ for j= 1, . . . , N. Suppose there exists b > 0 such that
ρ(BjBj+1· · · BN) ≤ b and ρ(Aj, Bj) ≤ b for j= 1, . . . , N.
Then there exists Cb> 0, depending on b, such that
ρ A1A2· · · AN, B1B2· · · BN ≤ Cb N X j=1 ρ(Aj, Bj)1/ν. (3.1)
The finite dimensional version of Lemma 3.1 can be found in [26, Lemma 3.7]. The proof works in a similar way for Banach homogeneous groups. However, due to the key role played by Lemma 3.1 in our arguments, we will present its proof in Section 5.2. Since any stratified group is a Banach homogeneous group, we can also apply Lemma 3.1 in , replacing ν by s and ρ by d.
We will also need the following estimate for distances in stratified groups [15, Lemma 2.13]. Lemma 3.2. There is a constant D> 0 such that
d(x−1yx) ≤ Dd(y)+ d(x)1sd(y) s−1 s + d(x) s−1 s d(y) 1 s for x, y ∈ .
Unless otherwise stated, we denote by f : A → a fixed Lipschitz function on a measurable set A ⊂ with Lipschitz constant L.
Definition 3.3. Fix ζ, η ∈ , y, z ∈ and ε, δ > 0. Let C1 and C2 respectively be the
con-stants obtained from applying Lemma 3.1 in with b = max{ε, ρ(y) + ρ(z)} and in with b= max{ε, ε/L, d(ζ) + d(η)}.
We define P(ζ, η, y, z, ε, δ, A) to be the set of points p ∈ A for which the following properties hold: • For all 0 < t < δ there exist ζt
p, ηtp∈ satisfying (3.2) pζtp∈ A and d(ζ t p, δtζ) < εt, (3.3) pηtp∈ A and d(η t p, δtη) < εt, (3.4) ρ( f (p)−1f(pζtp), δty) ≤ εt, (3.5) ρ( f (p)−1f(pηtp), δtz) ≤ εt.
• For arbitrarily small t there exist ωtp∈ such that
(3.6) pωtp∈ A and d(ωtp, δt(ζη)) < εt, (3.7) ρ( f (p)−1f(pωtp), δt(yz)) > 3C1ε1/ν+ Lε + LC2 2+ L−1/s ε1/st.
Intuitively, P(ζ, η, y, z, ε, δ, A) consists of points for which the directional derivatives in direction ζ and η look like y and z respectively, at scales less than δ and with accuracy ε.
Notice that provided ε and δ are bounded by some fixed constant K, the constants C1 and C2in
Definition 3.3 are bounded by constants independent of the precise value of ε and δ. Lemma 3.4. The set P(ζ, η, y, z, ε, δ, A) is porous.
Proof. For this proof we abbreviate P = P(ζ, η, y, z, ε, δ, A). Let x ∈ P and choose 0 < t < δ/2 for which there exist ωtx ∈ satisfying (3.6) and (3.7). Fix also ζxt satisfying (3.2) and (3.4). Since x
was any element of P and t could be chosen arbitrarily small, to prove P is porous it suffices to show that B(xζtx, εt/L) ∩ P = ∅.
We suppose B(xζxt, εt/L) ∩ P , ∅ and deduce a contradiction. Choose p ∈ B(xζtx, εt/L) ∩ P. Use
the definition of P to choose ηtp∈ satisfying (3.3) and (3.5). We first estimate
ρ( f (x)−1f(pηt
p), δt(yz)).
To this end, observe we can write
where
A1= δ1/t( f (x)−1f(xζtx)), A2 = δ1/t( f (xζtx)−1f(p)), A3= δ1/t( f (p)−1f(pηtp)),
and
B1= y, B2= 0, B3= z.
We check Ai, Bi satisfy the hypotheses of Lemma 3.1 with b = max{ε, ρ(y) + ρ(z)}. Using (3.4)
gives ρ(A1, B1)= ρ(δ1/t( f (x)−1f(xζtx)), y)= 1 tρ( f (x) −1f(xζt x), δty) ≤ ε.
Recalling that p ∈ B(xζtx, εt/L), we obtain
ρ(A2, B2)= ρ(δ1/t( f (xζtx)−1f(p)), 0)= 1 tρ( f (p), f (xζ t x)) ≤ L td(xζ t x, p) ≤ ε.
Using (3.5) also gives
ρ(A3, B3)= ρ(δ1/t( f (p)−1f(pηtp)), z)=
1 tρ( f (p)
−1f(pηt
p), δtz) ≤ ε.
Hence ρ(Ai, Bi) ≤ ε for each i. Clearly also ρ(BiBi+1· · · B3) ≤ ρ(y)+ ρ(z) for each i. Hence the
hypotheses of Lemma 3.1 are satisfied for our choice of b and we get (3.8) ρ( f (x)−1f(pηtp), δt(yz)) ≤ C1t
3
X
i=1
d(Ai, Bi)1/ν≤ 3C1ε1/νt.
Now let θtx = x−1pηtp. Then xθtx = pηtp ∈ A and (3.8) gives
(3.9) ρ( f (x)−1f(xθtx), δt(yz)) ≤ 3C1ε1/νt.
Now we estimate d(θtx, δt(ζη)). Since p ∈ B(xζtx, εt/L), we may choose ht ∈ with d(ht) ≤ εt/L
and p= xζtxht. Therefore d(θtx, δt(ζη))= d(x−1pηtp, δt(ζη))= d(ζtxh tηt p, δt(ζη)). Now define A1= δ1/tζtx, A2= δ1/tht, A3= δ1/tηtp, and B1 = ζ, B2 = 0, B3= η.
Let b= max{ε, ε/L, d(ζ) + d(η)}. Using (3.2) and (3.3) shows d(A1, B1) and d(A3, B3) are bounded
by ε. Using d(ht) ≤ εt/L gives d(A2, B2) ≤ ε/L. Clearly d(BiBi+1· · · B3) ≤ d(ζ)+ d(η) for each i.
Hence the hypotheses of Lemma 3.1 are satisfied, giving d(θtx, δt(ζη)) ≤ C2t 3 X i=1 ρ(Ai, Bi)1/s ≤ C2 2+ L−1/s ε1/st.
Recall ωtx were chosen satisfying (3.6) and (3.7). Using also (3.9), we estimate as follows: ρ( f (x)−1f(xωt x), δt(yz)) ≤ ρ( f (x)−1f(xθtx), δt(yz))+ ρ( f (xωtx), f (xθ t x)) ≤ 3C1ε1/νt+ Ld(ωtx, θtx) ≤ 3C1ε1/νt+ L d(ωtx, δt(ζη))+ d(θtx, δt(ζη)) ≤ 3C1ε1/νt+ Lεt + LC2 2+ L−1/s ε1/st = 3C1ε1/ν+ Lε + LC2 2+ L−1/s ε1/st.
This contradicts the choice of ωtx, forcing us to conclude B(xζtx, εt/L) ∩ P = ∅. Hence P is porous,
which concludes the proof.
We now prove Theorem 1.2 by putting together countably many of the sets from Definition 3.3. Proof of Theorem 1.2. Using Theorem 5.2 we can assume that the Banach homogeneous group is separable. Let P be the countable union of sets P(ζ, η, y, z, ε, δ, A) for ζ, η in a countable dense subset Nof , y, z in a countable dense subset Nof , and ε, δ positive rationals. By Proposition 2.10,
it is enough to show that for points x ∈ A ∩ D(A) \ P we have ∂+f(x, ζη)= ∂+f(x, ζ)∂+f(x, η) whenever ζ, η ∈ for which ∂+f(x, ζ), ∂+f(x, η) exist.
Suppose x ∈ A ∩ D(A) \ P and ∂+f(x, ζ), ∂+f(x, η) exist for some ζ, η ∈ . Fix ε > 0 rational. Choose ζxt, ηtx∈ for t > 0 with xζtx, xηtx∈ A and d(ζxt, δtζ)/t, d(ηtx, δtη)/t → 0. Using the existence
of ∂+f(x, ζ), ∂+f(x, η), choose δ > 0 rational such that for 0 < t < δ:
d(ζtx, δtζ) < εt/2 and ρ(δ1/t( f (x)−1f(xζxt)), ∂+f(x, ζ)) ≤ ε/2,
d(ηtx, δtη) < εt/2 and ρ(δ1/t( f (x)−1f(xηtx)), ∂+f(x, η)) ≤ ε/2.
Now choose y, z ∈ Nsuch that
(3.10) ρ(∂+f(x, ζ), y) < ε/2, ρ(∂+f(x, η), z) < ε/2, ρ(∂+f(x, ζ) ∂+f(x, η), yz) < C1ε1/ν.
In particular, we have
(3.11) ρ(δ1/t( f (x)−1f(xζtx)), y) ≤ ε and ρ(δ1/t( f (x)−1f(xηtx)), z) ≤ ε for 0 < t < δ.
Now choose ζ0, η0 ∈ Nsuch that d(ζ, ζ0) < ε/2, d(η, η0) < ε/2 and d(ζη, ζ0η0) < ε/2. Then
(3.12) d(ζxt, δtζ0) < εt and d(ηtx, δtη0) < εt for 0 < t < δ.
Equations (3.11) and (3.12) show that the point x satisfies the first four conditions of the set P(ζ0, η0, y, z, ε, δ, A), i.e. (3.2)–(3.5) hold with ζ, η replaced by ζ0, η0and p replaced by x.
Since x < P ⊃ P(ζ0, η0, y, z, ε, δ, A), the analogue of (3.6) or (3.7) for the set P(ζ0, η0, y, z, ε, δ, A) must fail for sufficiently small t. Hence whenever t is sufficiently small and ωtx satisfy the two
conditions
(3.13) xωtx∈ A and d(ω
t
x, δt(ζ0η0)) < εt,
then we must have
(3.14) ρ( f (x)−1f(xωtx), δt(yz)) ≤
3C1ε1/ν+ Lε + LC2
Let ωtxapproximate in A the direction ζη at x. Then xωtx ∈ A and d(ωtx, δt(ζη)) < εt/2 for sufficiently
small t. Since d(ζη, ζ0η0) < ε/2 it follows that ωtx satisfy (3.13) and hence (3.14) for sufficiently
small t. Taking into account the third inequality of (3.10), we obtain for sufficiently small t: (3.15) ρ( f (x)−1f(xωtx), δt(∂+f(x, ζ)∂+f(x, η))) ≤
4C1ε1/ν+ Lε + LC2
2+ L−1/s ε1/st. To summarize, if ωtx approximate the direction ζη at x then (3.15) holds for sufficiently small t.
Provided ε, δ < 1, the constants C1 and C2 are independent of the precise value of ε and δ. We
conclude that ∂+f(x, ζη) exists and is equal to ∂+f(x, ζ)∂+f(x, η). 4. From directional derivatives to differentiability
We now prove Theorem 1.3 which gives conditions for pointwise differentiability. Combining these results with those of Section 3 will yield Theorem 1.1.
Proof of Theorem 1.3. First we suppose f is differentiable at x ∈ A ∩ D(A). Then there exists an h-homomorphism Lx : → such that
ρ( f (x)−1f(xz), L
x(z))= o(d(z)) as z ∈ x−1A and z →0.
(4.1)
Let ζ ∈ . Using Lemma 2.8, choose ζt
xsuch that xζtx ∈ A and d(ζtx, δtζ)/t → 0 as t ↓ 0. Then (4.1)
gives ρ f(x)−1f(xζtx), Lx(ζxt) = o(d(ζ t x)) as t ↓ 0. (4.2)
Since d(ζtx, δtζ)/t → 0, there exists K = K(ζ) > 0 such that d(ζtx) < Kt for sufficiently small t.
Combining this with (4.2) gives lim
t↓0 ρδ1/t( f (x) −1f(xζt
x)), δ1/tLx(ζtx) = 0.
(4.3)
Since Lxis an h-homomorphism (and hence Lipschitz), we have
ρ(Lx(ζ), δ1/tLx(ζtx))= ρ(Lx(ζ), Lx(δ1/tζxt)) ≤ Lip(Lx)
d(δtζ, ζtx)
t → 0.
Combining this with (4.3) shows ∂+f(x, ζ) exists and is equal to Lx(ζ) for any ζ ∈ . Since Lxis an
h-homomorphism, the h-homomorphism property of directional derivatives follows.
For the converse statement, we assume ∂+f(x, ζ) exists for any ζ ∈ and Lx : → defined
by Lx(ζ) := ∂+f(x, ζ) is an h-homomorphism. For every ζ ∈ with d(ζ) ≤ 1, use Lemma 2.8 to
choose ζxt for t > 0 with xζtx ∈ A and d(ζxt, δtζ)/t → 0 as t ↓ 0 uniformly for ζ ∈ with d(ζ) ≤ 1.
We first show that
(4.4) δ1/t( f (x)−1f(xζtx)) → ∂+f(x, ζ) as t ↓ 0 uniformly for ζ ∈ with d(ζ) ≤ 1.
Let K= 1 + Lip(Lx)+ 3L > 0 and fix ε > 0. Choose a finite set S ⊂ such that for any η ∈
with d(η) ≤ 1, there exists ζ ∈ S with d(ζ, η) < ε/K. Choose δ > 0 such that
ρδ1/t( f (x)−1f(xζxt)), ∂+f(x, ζ) < ε/K for every ζ ∈ S and 0 < t < δ,
(4.5) and
d(ηtx, δtη)/t < ε/K for every η ∈ with d(η) ≤ 1 and 0 < t < δ.
Given η ∈ with d(η) ≤ 1, choose ζ ∈ S with d(ζ, η) < ε/K. We observe that ρδ1/t( f (x)−1f(xηtx)), ∂+f(x, η) ≤ρδ1/t( f (x)−1f(xζtx)), ∂+f(x, ζ) (4.7) + ρδ1/t( f (x)−1f(xηtx)), δ1/t( f (x)−1f(xζtx)) + ρ ∂+f(x, ζ), ∂+f(x, η).
The first term in (4.7) is estimated by (4.5) for 0 < t < δ. We estimate the second term of (4.7) as follows: ρδ1/t( f (x)−1f(xηtx)), δ1/t( f (x)−1f(xζtx)) = 1 tρ f(xηtx), f (xζ t x) ≤ L td(η t x, ζ t x) ≤ L d(η t x, δtη) t + d(ζxt, δtζ) t + d(ζ, η) ! . Using our choice of ζ and (4.6), we obtain
(4.8) ρδ1/t( f (x)−1f(xηtx)), δ1/t( f (x)−1f(xζxt)) < 3Lε/K for 0 < t < δ.
We estimate the final term in (4.7) as follows:
ρ(∂+f(x, ζ), ∂+f(x, η))= ρ(Lx(ζ), Lx(η)) ≤ Lip(Lx)d(ζ, η) < εLip(Lx)/K.
(4.9)
Using (4.5), (4.8) and (4.9) in (4.7) together with the definition of K yields ρδ1/t( f (x)−1f(xηtx)), ∂+f(x, η) < ε.
This proves the uniform convergence (4.4).
To conclude we will show that ρ( f (x)−1f(xz), Lx(z)) = o(d(z)) as z → 0 with z ∈ x−1A. First
notice that for ζ ∈ with d(ζ) ≤ 1: 1 tρ( f (x) −1f(xζt x), Lx(ζxt))= ρ(δ1/t( f (x)−1f(xζxt)), δ1/tLx(ζtx)) ≤ρ(δ1/t( f (x)−1f(xζxt)), ∂+f(x, ζ))+ ρ(∂+f(x, ζ), δ1/tLx(ζtx)) = ρ(δ1/t( f (x)−1f(xζxt)), ∂+f(x, ζ))+ ρ(Lx(ζ), Lx(δ1/tζxt)) ≤ρ(δ1/t( f (x)−1f(xζxt)), ∂+f(x, ζ))+ Lip(Lx)d(ζ, δ1/tζtx).
Hence (4.4) and our choice of ζtxgives
1 tρ( f (x)
−1f(xζt
x), Lx(ζxt)) → 0 as t ↓ 0 uniformly for d(ζ) ≤ 1.
(4.10)
Given ε > 0, use (4.10) and our choice of ζtxto choose δ > 0 such that
d(δtζ, ζtx)
t < ε and
ρ( f (x)−1f(xζt
x), Lx(ζxt))
Now let z ∈ x−1Awith d(z) ≤ δ. Choose ζ ∈ with d(ζ) = 1 such that z = δtζ for some 0 < t < δ.
We then estimate as follows: ρ( f (x)−1f(xz), L x(z)) d(z) ≤ ρ( f (x)−1f(xδ tζ), f (x)−1f(xζxt)) t + ρ( f (x)−1f(xζt x), Lx(ζtx)) t + ρ(Lx(ζxt), Lx(δtζ)) t ≤ (L+ Lip(Lx)) d(δtζ, ζtx) t + ρ( f (x)−1f(xζt x), Lx(ζtx)) t < (L + Lip(Lx)+ 1) ε.
This shows that f is differentiable at x.
Using our results above, we can now quickly show how Theorem 1.3 gives Theorem 1.1. By [26, Lemma 4.9], there exists a spanning set of directions ν1, . . . , νN ∈ V1for some N ∈ N, namely there
exists T > 0 such that
δ
t1v1δt2v2· · ·δtNvN: 0 ≤ ti< T
contains the unit ball of . If H1⊂ has the RNP then Theorem 2.16 yields a null set Z ⊂ A such
that for every x ∈ A \ Z: A is dense at x in direction νiand ∂+f(x, νi) exists for every i= 1, . . . , N.
By Theorem 1.2, we can find a σ-porous set P ⊂ such that directional derivatives at x act as h-homomorphisms whenever x ∈ A ∩ D(A) \ P.
Combining these with the spanning property of v1, . . . , vN, it follows that f is differentiable at
each x ∈ A∩D(A)\(P∪Z) in any direction ζ of the unit ball. Proposition 2.10 extends this directional differentiability to all directions ζ of . The same Theorem 1.2 shows that for x ∈ A∩ D(A)\(P∪Z) the directional derivative 3 ζ → ∂+f(x, ζ) defines an h-homomorphism. We have established both conditions 1 and 2 of Theorem 1.3, therefore f is differentiable at every point of A ∩ D(A) \ (P ∪ Z). Since HdQ(A \ D(A))= 0 and porous sets have measure zero, our claim follows.
Remark 4.1. Taking into account Remark 2.12, when the domain A of f is an open set, properties (3.2) and (3.3) of Definition 3.3 are automatically satisfied. Hence several arguments in the proofs of both Theorem 1.2 and Theorem 1.3 become simpler.
5. Appendix
5.1. Separable Banach Homogeneous Groups. Let be a Banach homogeneous group of step ν. For every v = (v1, . . . , vk) ∈ k, k ≥ 1, we define the nonassociative product
pk(v)= [· · · [[v1, v2], v3] · · · ], vk]= v1◦ v2◦ · · · ◦ vk,
where pk : k → is clearly a continuous multilinear mapping. For any countable subset
N =nxj ∈ : j ∈ No ,
we define its rational homogeneous span as follows hhN ii= n X j=0 ν X l=1 λjlπl(xj) : λjl∈ Q, n ∈ N ⊂ .
The closed homogeneous span of N is simply hNi= hhNii ⊂ . This is clearly a linear subspace of that is also homogeneous, that is δrhN i ⊂ hN i for every r > 0. Since hhNii is countable, hNi
is a separable homogeneous Banach space, that in particular contains N.
Theorem 5.1. There exists a separable Banach homogeneous subgroup ⊂ such that hN i ⊂ . Proof. Let 1= hNi be the separable and homogeneous Banach space spanned by N and consider
k = X v∈I pk(v) : I ⊂ k1is finite
for each k = 2, . . . , ν. Since dilations are also Lie algebra homomorphisms and 1 is closed under
dilations, it follows that δrk ⊂ k for each k = 1, . . . , ν and r > 0. It is clear that k is a linear
subspace also for k= 2, . . . , ν.
We wish to show that for any couple of finite subsets I ⊂ k1and J ⊂ l1with X v∈I pk(v) ∈ k and X w∈J pl(w) ∈ l
the following conditions hold (5.1) y,X w∈J pl(w) ∈ l+1 and X v∈I pk(v), X w∈J pl(w) ∈ k+l
where y ∈ 1. Notice that the nilpotence of gives k+l = {0} whenever k + l > ν. The first
condition of (5.1) is straightforward: y,X v∈J pl(w) = − X v∈J pl+1((w, y)) ∈ l+1.
The second one for k= 2 is a consequence of the Jacobi identity, that yields [v1, v2], X w∈J pl(w) = − v2, X w∈J pl(w) , v1 − X w∈J pl(w), v1 , v2 = X w∈J pl+2((w, v2, v1)) − X w∈J pl+2((w, v1, v2)) ∈ l+2
for every positive integer l. If we assume the second condition of (5.1) to hold for a fixed k ≥ 2 and every l ≥ 1, then setting
v= (v1, v2, . . . , vk+1) ∈ k1+1 and ˜v= (v1, v2, . . . , vk) ∈ k1,
we consider the product pk+1(v), X w∈J pl(w) = [pk(˜v), vk+1], X w∈J pl(w) = − vk+1, X w∈J pl(w) , pk(˜v) − X w∈J pl(w), pk(˜v) , vk+1 = − pk(˜v), X w∈J pl+1((w, vk+1)) + X w∈J [[pk(˜v), pl(w)], vk+1] .
All addends of the previous sum belong to k+l+1by the inductive assumption, hence establishing
conditions (5.1). The linear subspace
V = 1+ 2+ · · · + ν ⊂
is closed under dilations, since so are all the subspaces j. Conditions (5.1) immediately show that
Vis a Lie subalgebra of . It turns out that its closure = V
is a Banach Lie subalgebra, that is also closed under dilations, due to the continuity of the Lie product and of dilations. Then the same argument in the proof of [26, Proposition 7.2] shows that is actually a direct sum of subspaces U1, . . . , Uν with [Ui, Uj] ⊂ Ui+ jand Ui+ j = {0} whenever
i+ j > ν. Since the group operation is given by the Baker-Campbell-Hausdorff formula, it turns out that is a Banach homogeneous subgroup of .
To show that is also separable, we first consider the countable subsets hhN iik = X v∈I pk(v) : I ⊂ hhNiik is finite where 2 ≤ k ≤ ν and define the closed subset
= hhNii + hhNii2+ · · · + hhNiiν.
Clearly is separable, being all the addends hhNii and hhNiik countable. The simple inclusion
hhN iik ⊂ kfor k ≥ 2 joined with 1= hhNii immediately lead us to the following
hhN ii+ hhNii2+ · · · + hhNνiiν⊂ V,
that gives ⊂ .
The opposite inclusion follows from the following
(5.2) ⊃V.
To prove this, we consider any element w= v + ν X k=2 X v∈Ik pk(v) ∈ V,
with v ∈ 1and Ik finite subset of k1. Since hhNii= 1, we find a sequence {yn} of hhNii with
yn → v. For k ≥ 2 and any v ∈ Ik we find a sequence {vn} ⊂ (hhNii)k such that vn → v. By the
continuity of p, we get pk(vn) → pk(v) as n → ∞. We conclude that zn = yn+ ν X k=2 X v∈Ik
pk(vn) ∈ hhNii+ hhNii2+ · · · + hhNiiν
and zn→ w ∈ , showing the validity of (5.2) and then concluding the proof.
Theorem 5.2. If A ⊂ and f : A → is continuous, then there exists a separable Banach homogeneous group 0⊂ such that f (A) ⊂ 0.
Proof. Let N ⊂ be a countable subset such that f (A) ⊂ N. Then Theorem 5.1 provides us with a separable Banach homogeneous group 0⊂ containing N, therefore concluding the proof.
5.2. Proof of Lemma 3.1. Fix b > 0. Since each map πi is continuous and linear, the following
inequality holds for |x| ≤ b, where c > 0 only depends on : |πi(x)|1/i≤ c1/i|x|1/i= c1/i(1+ b)1/i
|x| 1+ b !1/i (5.3) ≤ c1/i(1+ b)1i− 1 ν|x|1/ν ≤ max{c, 1} (1+ b)1−1ν |x|1/ν, By (2.3) and (5.3) there exists C1,b> 0, depending on b, such that
(5.4) kxk ≤ C1,b|x|1/ν for |x| ≤ b.
Conversely, from (2.3) one may easily see that there exists C2,b > 0 such that
|x| ≤ C2,b for kxk ≤ b.
(5.5)
The same formula also yields (5.6) |x| ≤ ν X i=1 |πi(x)| ≤ max 1≤i≤ν ( 1 (σi)i ) ν X i=1 C1,bb1/ν i−1 kxk= C3,bkxk for |x| ≤ b.
We observe that for the addends in (2.2) to be non-zero, the iterated nonassociative product must always start from the factor x ◦ y or y ◦ x. Thus, the continuity of the Lie bracket yields
(5.7) |x ◦ · · · ◦ x | {z } p1times ◦ q1times z }| { y ◦ · · · ◦ y ◦ · · · ◦ x ◦ · · · ◦ x | {z } pktimes ◦ qktimes z }| { y ◦ · · · ◦ y| ≤ Cm−2bm−2|[x, y]| wherePk
i=1(pi+ qi)= m and we have assumed in addition that |x|, |y| ≤ b for some b > 0.
Claim 1. Let b> 0. Then there exists C4,b> 0, depending on b, such that
ky−1xyk ≤ C4,b|x|1/ν for |x|, |y| ≤ b. (5.8)
Proof. Recalling y−1= −y and applying Dynkin’s formula with the pairs −y, xy then x, y, we obtain y−1xy= x + ν X m=2 Pm(x, y)+ ν X m=2 Pm(−y, xy). (5.9)
Taking into account (5.7), it is easy to see that there exists C5,b > 0, such that
|y−1xy| ≤ C5,b|x|+ ν X m=2 Pm(−y, xy) .
Another application of (5.7), joined with (2.2), gives a constant C6,b> 0 such that
|xy| ≤ C6,b for |x|, |y| ≤ b.
Again using (5.7), it follows that
|Pm(−y, xy)| ≤ C7,b|[y, xy]|
(5.10)
for each m= 2, . . . , ν and a suitable constant C7,b > 0, depending on b. Thus, we have proved the
existence of C8,b> 0, depending on b, such that
Now, we observe that
[y, xy]= y, x +
ν X m=2 Pm(x, y)= [y, x] + y, ν X m=2 Pm(x, y).
This together with the continuity of [·, ·] leads us to the estimate |[y, xy]| ≤ C0 |x||y|+ |y| ν X m=2 |Pm(x, y)|
where C0 > 0 only depends on the Banach norm chosen on . As a consequence, there exists
C9,b> 0 such that
(5.12) |[y, xy]| ≤ C9,b|x| for |x|, |y| ≤ b.
The previous inequality joint with (5.11) gives |y−1xy| ≤ C10,b for some C10,b > 0 depending on b.
This bound on |y−1xy|allows us to apply (5.4), that joined with (5.11) and (5.12) gives (5.8). Claim 2. Let N be a positive integer and let Aj, Bj ∈ with j = 1, . . . , N. Let b > 0 be such that
|BjBj+1· · · BN| ≤ b and |B−1j Aj| ≤ b for every j= 1, . . . , N. Then there exists C > 0 such that
ρ A1A2· · · AN, B1B2· · · BN ≤ρ(AN, BN)+ C N−1 X j=1 |A−1j Bj|1/ν. (5.13)
Proof. Define ˆBj = BjBj+1· · · BN and ˆAj = AjAj+1· · · AN. Using the left invariance of ρ and
Claim 1, we obtain
ρ ˆA1, ˆB1= ρ ˆA2, A−11 B1Bˆ2 ≤ρ( ˆA2, ˆB2)+ k ˆB−12 A −1 1 B1Bˆ2k ≤ρ( ˆA2, ˆB2)+ C4,b|A−11 B1|1/ν ≤ρ( ˆA3, ˆB3)+ C4,b|A−12 B2|1/ν+ C4,b|A−11 B1|1/ν ≤ρ(AN, BN)+ C N−1 X j=1 |A−1j Bj|1/ν.
From the hypotheses of Lemma 3.1, taking into account (5.5), we see
|BjBj+1· · · BN| ≤ C2,b and |A−1j Bj| ≤ C2,b for j= 1, . . . , N.
Hence we may apply Claim 2. Recalling that ρ(AN, BN) ≤ b and using (5.6) we obtain
ρ A1A2· · · AN, B1B2· · · BN ≤ρ(AN, BN)+ C3,C2,b N−1 X j=1 kA−1j Bjk1/ν ≤ Cb N X j=1 ρ(Aj, Bj)1/ν, where Cb= max n b1−1ν, C3,C 2,b o
. This concludes the proof of Lemma 3.1.
5.3. Measurability of points where a directional derivative exists. The aim of this section is to prove that the set Df,ζ in the proof of Theorem 2.16 is measurable. Recall that f : A → is
Lipschitz and A ⊂ is measurable. It was shown earlier that the set Aζ of points at which A is dense in direction ζ is measurable and µ(A \ Aζ)= 0. The set Df,ζis the subset of points in Aζ where f is differentiable in direction ζ. By Theorem 5.2, we can assume that the target is a separable
Banach homogeneous group. Fix zi ∈ with {zi: i ∈ N} = . Let Pi(z) = ρ(z, zi) for z ∈ . For
y ∈ Aand t > 0, define measurable functions git,y,+, git,y,−: Aζ → R ∪ {+∞, −∞} by:
git,y,±(x)= ( Piδ1/t f(x)−1f(y) if y ∈ A ∩ B(xδtζ, t2) ±∞ if y ∈ A \ B(xδtζ, t2).
Fix a countable set S ⊂ A with A ⊂ S. Define x →ϕ(x) = lim
R→+∞supi∈N 0<s,t<1/Rsup s,t∈Q
sup
(y,z)∈S2
git,y,−(x) − gis,z,+(x) ,
which is well defined and measurable on Aζ. We claim that f is differentiable at x in direction ζ if and only if ϕ(x)= 0.
First suppose x ∈ Aζand f is differentiable at x in direction ζ. Proposition 2.14 implies that lim
t↓0, xδtζ∈A δ1/t
f(x)−1f(xδtζ) = ∂+f(x, ζ).
Since x is a density point of A in direction ζ, there exists Tt > 0 with xδTtζ ∈ A and Tt/t → 1 as t ↓0. For y ∈ B(xδtζ, t2), let
Ex,y,t= δ1/t
f(xδTtζ)
−1f(y) Taking into account that ζ is horizontal, we have
ρ(Ex,y,t) ≤ Ld(xδTtζ, y) t < L t |Tt− t| d(ζ)+ t2 = Lθt, where θt → 0 as t ↓ 0. We set (δ f )x,ζ,t= δ1/t f(x)−1f(xδTtζ) ,
observing that (δ f )x,ζ,t→∂+f(x, ζ) as t ↓ 0. The following estimates hold Piδ1/t f(x)−1f(y)− Pi (δ f )x,ζ,t = Pi (δ f )x,ζ,tEx,y,t− Pi (δ f )x,ζ,t ≤ρ(Ex,y,t) < Lθt uniformly in y ∈ B(xδtζ, t2) and i ∈ N. Since P : → `∞, z → (Pi(z))i∈N is an isometric
embedding, we get sup i∈N Pi (δ f )x,ζ,t− Pi ∂+f(x, ζ) → 0 as t ↓ 0. If we fix ε > 0, then for some Rε> 0 we obtain
Piδ1/t f(x)−1f(y)− Pi ∂+f(x, ζ) < ε/2
for 0 < t < 1/Rε, i ∈ N and y ∈ B(xδtζ, t2). In particular, for y ∈ B(xδtζ, t2) and z ∈ B(xδsζ, s2), it
follows that Piδ1/t f(x)−1f(y)− Piδ1/s f(x)−1f(z) < ε
for 0 < t, s < 1/Rεand i ∈ N. The fact that A is dense at x in direction ζ implies that, up to taking a larger Rε, the following
sup 0<s,t<1/Rε s,t∈Q sup (y,z)∈S2 git,y,−(x) − gis,z,+(x)
exactly equals sup 0<s,t<1/Rε s,t∈Q sup (y,z)∈S2 y∈B(xδtζ,t2), z∈B(xδsζ,s2) Piδ1/t f(x)−1f(y)− Piδ1/s f(x)−1f(z) < ε
for every i ∈ N. This proves that ϕ(x) = 0.
Conversely, we assume that ϕ(x)= 0 and wish to prove that f is differentiable at x in direction ζ. Let ε > 0 be arbitrarily fixed and choose Rε> 0 such that
(5.14) sup i∈N sup 0<s,t<1/Rε s,t∈Q sup (y,z)∈S2 git,y,−(x) − gis,z,+(x) < ε.
Observing that in the expression (5.14) we can exchange (t, y) with (s, z), if y ∈ B(xδtζ, t2) and
z ∈ B(xδsζ, s2) we get Piδ1/t f(x)−1f(y)− Piδ1/s f(x)−1f(z) ≤ε
for 0 < s, t ≤ 1/Rε. Since x ∈ A is a density point in direction ζ, we may consider again Tt > 0
such that xδTtζ ∈ A and Tt/t → 1 as t ↓ 0. Fix δ0 > 0 such that Tt < 2t for 0 < t ≤ δ0. Whenever 0 < s, t < 1/2Rε, we have 0 < Ts, Tt < 1/Rεand then
(5.15) Piδ1/Tt f(x)−1f(xδTtζ) − Piδ1/Ts f(x)−1f(xδTsζ) ≤ε for all i ∈ N. In particular, there exists
li = lim
t↓0 Piδ1/Tt
f(x)−1f(xδTtζ) and we can pass to the limit in (5.15) with respect to s ↓ 0, that yields
Piδ1/Tt f(x)−1f(xδTtζ) − li ≤ε. In particular, w= (li) ∈ `∞and kPδ1/Tt f(x)−1f(xδTtζ) − wk`∞ → 0 as t ↓ 0. Hence the following limit exists
lim
t↓0 δ1/Tt
f(x)−1f(xδTtζ) . By Remark 2.11, this shows that f is differentiable at x in direction ζ.
We conclude that Df,ζ =nx ∈ Aζ : ϕ(x)= 0o, from which the measurability of ϕ gives our claim. References
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(Valentino Magnani) Department of Mathematics, University of Pisa, Largo Pontecorvo 5, 56127 Pisa, Italy E-mail address: valentino.magnani@unipi.it
(Andrea Pinamonti) Department of Mathematics, University of Trento, Via Sommarive 14, 38123 Povo (Trento), Italy
E-mail address: andrea.pinamonti@unitn.it
(Gareth Speight) Department of Mathematical Sciences, University of Cincinnati, 2815 Commons Way, Cincinnati, OH 45221, United States
